Reflection, Refraction, and Snell's Law — The Complete Physics Guide
Optics Puzzle runs a full recursive ray-tracing engine — every mirror bounce, every refraction through glass, and every lens deflection is computed live using the same equations that govern real light. Placing mirrors and prisms to steer a laser beam to its target isn't a stylised abstraction of optics; it's the genuine physics, calculated the same way professional optical design software calculates it, just simplified to two dimensions.
The Law of Reflection
When light strikes a mirror, the angle of incidence exactly equals the angle of reflection, both measured from the normal — an imaginary line perpendicular to the surface at the point of contact. The incident ray, the reflected ray, and the normal all lie in the same plane. This single rule governs every mirror bounce in this game, and it applies not just to light but to any wave, including sound and radar.
Curved mirrors extend this same law to more useful effect. A concave parabolic mirror reflects every ray parallel to its axis so that they all converge at a single focal point, located at f = R/2 from the mirror's surface, where R is its radius of curvature. This one geometric fact is the working principle behind reflecting telescopes, satellite dishes, and the parabolic reflectors behind car headlights and torch bulbs.
Snell's Law and Refraction
When light crosses a boundary between two materials of different refractive index, it bends — a phenomenon called refraction, governed by Snell's law: n₁ sinθ₁ = n₂ sinθ₂. The refractive index n of a material is defined as n = c/v, the ratio of light's speed in a vacuum to its speed inside that material. Typical values: air ≈ 1.000, water = 1.333, crown glass ≈ 1.52, and diamond a striking 2.42.
Light always bends toward the normal when entering a denser medium (higher n) and away from the normal when exiting into a less dense one. The underlying reason is that light is a wave, and one edge of its wavefront enters the new medium — and changes speed — before the rest of it does, rotating the wavefront's direction of travel exactly the way a car's front wheels turn when one side hits mud before the other.
This bending is responsible for a whole family of everyday optical illusions: a swimming pool looking shallower than it really is, a straw appearing to bend sharply at the water's surface, and desert mirages that look like distant pools of water — all consequences of light refracting through media (or layers of air) with different refractive indices.
Total Internal Reflection and the Critical Angle
When light travels from a denser medium into a less dense one (n₁ > n₂) at a sufficiently steep angle, something remarkable happens: instead of refracting out, all of the light reflects back into the denser medium. This is total internal reflection, and it occurs whenever the angle of incidence exceeds a critical angle θ_c = arcsin(n₂/n₁).
For glass (n = 1.52) meeting air, the critical angle is 41.1°; for diamond, it's a remarkably small 24.4°. That small critical angle means light entering a diamond is very likely to strike an internal facet at an angle beyond it, triggering multiple total internal reflections before finally emerging — which is exactly why cut diamonds display such extraordinary brilliance and fire compared to other gemstones with less extreme refractive indices.
Total internal reflection is also the entire operating principle behind optical fibres. A glass fibre core (n ≈ 1.5) is surrounded by a cladding layer of slightly lower refractive index (n ≈ 1.46), and light injected at a shallow enough angle reflects internally over and over as it travels — trapped inside the fibre for kilometres with minimal loss, enabling the terabit-per-second data transmission that underpins the modern internet.
Lenses and Focusing
A lens works by refracting light at two curved surfaces in sequence, and its overall effect on parallel rays is captured by the thin lens equation, 1/f = 1/v − 1/u, relating focal length f, image distance v, and object distance u. A convex (converging) lens brings parallel incoming rays together at a real focal point; a concave (diverging) lens spreads them apart as though they originated from a virtual focal point behind the lens.
Every optical instrument you've ever used — a camera, a pair of glasses, a microscope, a telescope's eyepiece — is built from combinations of these two basic lens behaviours, arranged to form, magnify, or correct an image in a specific, calculated way. The puzzles in this game that involve bending a beam through a lens are computing exactly this refraction, just isolated down to a single representative ray rather than the full cone of light a real lens handles.
Dispersion — Why Prisms Split White Light
Refractive index isn't quite a single fixed number for a given material — it varies slightly with the wavelength of light passing through, a phenomenon called dispersion. In most transparent materials, blue and violet light (shorter wavelengths) bend more than red light (longer wavelengths) when refracting, because they experience a slightly higher refractive index.
A glass prism exploits this directly: white light entering one face refracts, with each colour bending by a very slightly different amount, then refracts again on exiting the second face, further separating the colours until they emerge as a visible rainbow spectrum. Raindrops perform exactly the same trick on a larger, more complex scale — refracting, internally reflecting, and refracting again inside each droplet — which is why a rainbow's colours always appear in the same order, violet through red, whenever sunlight and rain combine at the right angle.
Worked Example — Finding the Critical Angle
Problem: Find the critical angle for light travelling from crown glass (n = 1.52) into air (n = 1.00), and determine whether a ray hitting the glass-air boundary at 45° undergoes total internal reflection.
θ_c = arcsin(n₂/n₁) = arcsin(1.00/1.52) = arcsin(0.658) ≈ 41.1°
Since 45° > 41.1°, the ray exceeds the critical angle — it undergoes total internal reflection and does not exit into the air at all.
Real-World Applications
Fibre-optic communication: The entire global internet backbone relies on total internal reflection in glass fibres to carry data as pulses of light across oceans and continents with minimal signal loss, at a fraction of the energy cost of equivalent electrical transmission.
Endoscopes and medical imaging: Bundles of optical fibres, each individually guiding light by total internal reflection, allow doctors to see inside the human body through incisions just millimetres wide, transforming diagnostic and minimally invasive surgical medicine.
Periscopes and binoculars: Many optical instruments use 45°-45°-90° prisms rather than silvered mirrors to redirect light, because the internal reflection off the prism's hypotenuse face is total (and therefore lossless) provided the light strikes it beyond the critical angle — a cheaper, more durable, and optically cleaner solution than a coated mirror.